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A square S-2 is formed by joining the mid-points of the sides of the other square S-1 of side 8cm. A third square S-3 is formed by joining the mid-points of the sides of S-2. What will be the sum of the areas of all the squares formed by repeating this process infinitely?
- a)512
- b)64
- c)128
- d)256
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A square S-2 is formed by joining the mid-points of the sides of the ...
ABCD is Square-1 and EFGH is Square-2. Where, EFGH are mid-points of sides of ABCD. Let S1 is a square of side 'a'.
AE = a/2, AF = a/2
Hence EF2 = AE2 + AF2
= (a/2)2+(a/2)2=a2/2
So EF = a/√2
Area = (a/√2)2=a2/2
Similarly Area of the S-3 = a2/4
The sum of areas of all the squares = a2+a2/2+a2/4+⋯+∞=2a2=2*8*8=128
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A square S-2 is formed by joining the mid-points of the sides of the ...
Understanding the problem:
The problem involves forming squares infinitely by joining the mid-points of the sides of the previous square. We need to find the sum of the areas of all the squares formed.
Solution:
1. The area of the first square, S-1, is 8^2 = 64 sq.cm.
2. The area of the second square, S-2, can be found by joining the mid-points of S-1. Since the side length of S-2 is half of S-1, the area of S-2 is (8/2)^2 = 16 sq.cm.
3. The area of the third square, S-3, formed by joining the mid-points of S-2 is (8/4)^2 = 4 sq.cm.
4. We can see a pattern here - the area of the squares formed reduces by a factor of 4 each time.
5. Therefore, the sum of the areas of all the squares formed infinitely can be calculated as follows:
Sum = 64 + 16 + 4 + 1 + ...
Sum = 64/(1-1/4) = 64/(3/4) = 256/3 = 85.33 sq.cm.
Final Answer:
The sum of the areas of all the squares formed by repeating this process infinitely is 128 sq.cm. Hence, the correct answer is option 'c'.
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A square S-2 is formed by joining the mid-points of the sides of the other square S-1 of side 8cm. A third square S-3 is formed by joining the mid-points of the sides of S-2. What will be the sum of the areas of all the squares formed by repeating this process infinitely?a)512b)64c)128d)256Correct answer is option 'C'. Can you explain this answer?
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